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Quantum Entanglement of a Harmonic Oscillator with an Electromagnetic Field

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www.nature.com/scientificreports OPEN Quantum entanglement of a harmonic oscillator with an electromagnetic feld Received: 23 November 2017 Dmitry N. Makarov Accepted: 2 May 2018 At present, there are many methods for obtaining quantum entanglement of particles with an Published: xx xx xxxx electromagnetic feld. Most methods have a low probability of quantum entanglement and not an exact theoretical apparatus based on an approximate solution of the Schrodinger equation. There is a need for new methods for obtaining quantum-entangled particles and mathematically accurate studies of such methods. In this paper, a quantum harmonic oscillator (for example, an electron in a magnetic feld) interacting with a quantized electromagnetic feld is considered. Based on the exact solution of the Schrodinger equation for this system, it is shown that for certain parameters there can be a large quantum entanglement between the electron and the electromagnetic feld. Quantum entanglement is analyzed on the basis of a mathematically exact expression for the Schmidt modes and the Von Neumann entropy. It is known that quantum entanglement arises when the wave function of a particle system cannot be represented as a product of the wave functions of each particle. Tere are many such systems from which to choose. Te basic method of quantum entanglement of particles is spontaneous parametric down-conversion (SPDC)1–3. In this process, when a nonlinear birefringent crystal is irradiated by a laser pump feld the pump photons decay with a low probability into two quantum-entangled photons of lower frequencies. Despite the low probability of the decay of the pump photon, SPDC has been widely studied and has many practical applications in quantum com- puter science. Tis is due to the fact that this method is the simplest for obtaining quantum-entangled photons. While other more sophisticated methods of generating entangling particles have been proposed4–8 none of them result in a greater probability of entanglement of the particles under consideration than SPDC9,10. Tere is also a difculty with the theoretical study of the various methods, which is that basically, quantum entanglement is a complex process for the mathematical research. For the systems under consideration, it is impossible to solve exactly the nonstationary Schrodinger equation and calculate the entanglement parameter (for example, the Von Neumann entropy in the case of a two-component system). Solutions are mainly sought using higher orders of perturbation theory and on the basis of these solutions conclusions are drawn about quantum entanglement. It is known that perturbation theory is applicable if there is a weak interaction of particles in the system, and therefore use of this theory is based on the assumption that quantum entanglement is not a large quantity. In order to study a system with large quantum entanglement, the exact solution of the Schrodinger equation must be used - a task that is very difcult to achieve mathematically. Nonetheless, when studying quantum entanglement, such a solu- tion would be of fundamental interest. In quantum physics, it is known that exact solutions of the Schrodinger equation do present any great difculty. In studying quantum entanglement, one can include a two-level model of Jaynes-Cummings11 in order to achieve sufciently accurate solutions. In this model are studied the physical characteristics of a two-level atom in an electromagnetic feld. It should also be noted that the Jaynes-Cummings model describes the system of a two-level atom interacting with a quantized electromagnetic feld, which greatly limits its application to many physical systems (the results presented in this work go beyond this limitation). Also some model problems and particular cases for the stationary Schrodinger equation12,13. Quantum Dynamics of quantum-entangled systems is being studied actively at the present time14–17. For example, on the basis of the exact solution of the nonstationary Schrodinger equation for an atom in a strong two-mode electromagnetic feld, it has been shown17,18 that there is a strong quantum entanglement between photons. However, as this solution is based upon the assumption that the external electromagnetic feld is many times stronger than the Coulomb feld of attraction of an electron in an atom, it does not promote an understanding of the infuence of the Coulomb Northern (Arctic) Federal University named after M.V. Lomonosov, Arkhangelsk, 163002, Russia. Correspondence and requests for materials should be addressed to D.N.M. (email: [email protected]) SCIENTIFIC REPORTS | (2018) 8:8204 | DOI:10.1038/s41598-018-26650-8 1 www.nature.com/scientificreports/ Figure 1. Interaction of an electromagnetic feld with an electron in a magnetic feld. feld on quantum entanglement. Terefore, the problem of fnding a system whose quantum entanglement is large, and also in which it would be possible to mathematically study quantum entanglement on the basis of an exact solution of the Schrodinger equation, is topical. In addition to this relevance, the problem of practical appli- cation is important, i.e. the proposed method should be simple enough for practical implementation. It should be added that quantum entanglement is widely used in quantum computer science. Tat is why at present a great interest is attracted to this problem. In particular, using entangled states, it is possible to explain quantum communication protocols19, such as quantum cryptography20, quantum coding21, quantum key distri- bution22, quantum secure direct communication23, quantum machine learning24,25, entanglement swapping26,27, quantum computing algorithms28 and quantum state teleportation29. Tere are quite a lot of practical problems in quantum informatics, one of which is the weak generation of quantum-entangled particles. Te method of generation of quanta-entangled particles proposed in this work gives a large amount of quantum entanglement. In this paper we study the generation method of quantum entanglement of a harmonic oscillator with an external electromagnetic feld. A harmonic oscillator can be an electron in a uniform magnetic feld with induc- tion B, see Fig. 1. It is known30 that in such a feld the electron behaves like a harmonic oscillator with frequency Be ωc = , where c is the speed of light, me and −e are, the mass and charge of an electron, respectively. Te wave mce function of such a system can be found on the basis of the exact solution of the Schrodinger equation of the spe- cifc system being studied. Te quantum entanglement of the system can be studied using the analytical form of the Schmidt mode31,32 and the von Neumann entropy33,34. It is shown that for certain parameters of the system under consideration, a large quantum entanglement is possible. Exact solution of the Schrodinger equation Let us consider, at a time t > 0, a quantized electromagnetic feld acting on a system consisting of a quantum harmonic oscillator. Te interaction of a harmonic oscillator of mass m occurs via an electric charge e. Te Schrodinger equation for this system will be 2 22 22 ∂Ψ 1 ∂ e mxωδc ((++yz)) i= −i + Aˆ ++Hˆ Ψ. f ∂tm2 ∂r c 2 (1) In the expression: (1) δ = 0 corresponds to one type of interaction, and δ = 1 to another type; i is the imaginary unit; ω is the frequency of the oscillator; Hˆ = ∑ ω aa+ + 1 is the Hamiltonian of the electromagnetic c f ku, ˆˆku,,ku 2 + () feld, where aˆku, and aˆku, are the creation and annihilation operators, respectively, for photons with the wave vec- tor k and the polarization u; and the vector potential of the electromagnetic feld Aˆ has the form 2 2πc ⁎ + Auˆ =−∑ ((expi()ωωtakr )(ˆˆku,,+−ukexpi()ta−.r ))ku ku, ωVf (2) Te summation for Hˆ f and expression (2) include all possible values of the wave vector k and the polarization u. Next we consider a single-mode feld, in which ∑ku, has one summand with the wave vector k with polarization u. Further, for convenience we apply a system of units similar to atomic ħ = 1, m = 1, e = 1. It should be added that any suitable physical system can be an oscillator. In such a system, the charge e and the mass m depends on the SCIENTIFIC REPORTS | (2018) 8:8204 | DOI:10.1038/s41598-018-26650-8 2 www.nature.com/scientificreports/ problem under consideration. Terefore, everywhere below, under the oscillator interacting with the electromag- netic feld, we mean any suitable physical system, the Schrodinger equation of which is described by the expres- sion (1). Now consider expression (1) in the dipole approximation in which the vector potential, expressed in 2 terms of feld variables, is equal to Auˆ = aq, where a = 4πc , ω is the frequency of the mode under considera- ωV tion, V is the quantization volume, and q is the feld variable of the mode. Since we are considering a single-mode electromagnetic feld, it is convenient to orient the polarization vector along the x axis i.e. u = i, where i is the unit vector directed along the x axis (the same assumption applies throughout the paper). For example, expression (1) for δ = 0 corresponds to the case of an electron in a magnetic feld interacting with an external quantized electromagnetic feld if the polarization of the electromagnetic feld u is in the plane perpendicular to the magnetic induction vector B. Tis is easily obtained by using the Landau gauge30 for the magnetic feld AL = Bxj, where j is the unit vector directed along the y axis. As a result, the dynamics of the system, and electronic transitions are not determined by the stationary Schrodinger equation with the Hamiltonian 2 1 ∂ ω ∂2 ω 22x Hiˆ = − + βqq + 2 − + c , 2 22 ∂x ∂q 2 (3) where β = 4π . It should be clarifed that expression (3) corresponds to the Hamiltonian without the variables ωV y, z, since the wave function corresponding to the solution of equation (1) Ψ(x, y, z, q, t) = Ψ(y, z, t)Ψ(x, q, t), and this means that Ψ(y, z, t) is not a perturbed wave function and therefore does not afect the electronic transitions of the system under the action of an external electromagnetic field.
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